How to make a fullscreen toggle button in the BGE?
This is simply done with a small python script using the bge.render.setFullScreen() function.

cont = bge.logic.getCurrentController()

if cont.sensors['MouseOver'].hitObject.name == 'Cube' and cont.sensors['Lclick'].getButtonStatus(bge.events.LEFTMOUSE) == bge.logic.KX_INPUT_JUST_ACTIVATED:
    if bge.render.getFullScreen() == True:
        bge.render.setFullScreen(False)
    else:
        bge.render.setFullScreen(True)


You then add the script to your button with these logic bricks.


The MouseOver and Lclick mouse sensors are used to start the script when the button is clicked.
Look through the Blend file for a full working example.
(*Note* you will need to start the Standalone player for full screen to work.)